Questions about the branch of abstract algebra that deals with groups.

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6
votes
0answers
182 views

Grothendieck - A group as a sheaf over simplicial complexes

In this blog post, Terence Tao gives the following definition of a group. Definition. A group is (identifiable with) a (set-valued) sheaf on the category of simplicial complexes such that the ...
19
votes
1answer
634 views

Why can the general quintic be transformed to $w^5-5\beta w^3+10\beta^2w-\beta^2 = 0$?

The quintic can be transformed to the one-parameter Brioschi quintic, $$w^5-10\alpha w^3+45\alpha^2w-\alpha^2 = 0\tag1$$ This form is well-known for its connection to the symmetries of the ...
7
votes
0answers
248 views

Pedagogical question on Lie groups vs. matrix Lie groups

There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...
-1
votes
1answer
82 views

Group associated to the monoid $({\cal P}(X\times X), \circ)$

Consider the set ${\cal P}(X\times X)$. It can be endowed with a binary operation $\circ$ where $$A\circ B = \{(a,b)\in X\times X:\exists z\in X((a,z)\in A\land (z,b)\in B)\}.$$ Note that ...
1
vote
1answer
50 views

How to claculate the $T$-stable subgroup of second cohomology group

Let $G=\langle x,y,z,w \mid [y,x]=w^p=z, x^{p^2}=y^p=z^p=1 \rangle$ be a group, where $[u,v]=u^{-1}v^{-1}uv$, $p$ is a prime and the commutator which do not appear is 1. Let $N=\langle y,w \rangle ...
3
votes
1answer
218 views

Finite groups of order $n$ having exactly $n$ subgroups

Is it known a characterization of finite groups of order $n$ having exactly $n$ subgroups? A supplementary question: are there abelian groups other than the trivial group and $\mathbb{Z}_2$ with ...
5
votes
1answer
105 views

Generalization of a lemma of Livne

Let $H$ be a finite $2$-group. Let $N_{4}(H)$ be the subgroup generated by fourth powers. Let $H_{4}$ be the last term in the short exact sequence $1\rightarrow N_4(H) \rightarrow H \rightarrow H_{4} ...
17
votes
2answers
654 views

How large can the smallest generating set of a group $G$ of order $n$ be?

Let $n$ be a natural number. For every group $G$ of order $n$, denote $d(G)$ : The number of elements of the smallest generating set of $G$ How large is the maximum possible value of $d(G)$ ...
13
votes
3answers
412 views

Explicit construction of an element of ${\rm GL}(2, p)$ of order $p+1$

It is well-known that the order of $GL(2, p)$ is $(p^2-1)(p^2-p) = (p-1)^2(p+1)p.$ It is easy to construct matrices of orders $(p-1)$ and $p$ (diagonal and parabolic, respectively), but the only way ...
0
votes
0answers
81 views

Is there a metric defined on the product space of orthogonal groups?

If one considers just the orthogonal group, then there is a natural metric given by [1]: $$\begin{align} \theta & =\frac{1}{2} \| \log(R_1^{-1}R_2) \| \\ & = \frac{1}{2} \| ...
9
votes
1answer
405 views

Dessins d'enfants and absolute Galois group

I would like to know what is the recent progress about the group homomorphism $$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$$ ...
5
votes
0answers
70 views

Some questions about the Lévy monoid of certain densities

Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$. Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...
1
vote
1answer
139 views

The groups with nilpotent Hall $p'$ subgroup

Theorem $1$(Burnside): A simple nonabelian finite group can not have a conjugacy classes with prime power elements. Theorem $2$: A group of order $p^nq^m$ is solvable. Theorem $1$ depends on ...
3
votes
0answers
113 views

Short exact sequences for amalgamated free products and HNN Extensions

I asked this question on math stackexchange (see here) but didn't get any answer so I thought I would post it here too: If $A$ and $B$ are groups we have the following short exact sequence: $$ 0 \to ...
4
votes
1answer
108 views

Genericity of irreducible automorphisms of free groups

I have seen in the literature that Irreducible outer automorphisms of a free group $F_n$ are "generic". I would like to ask that if for example : it is true that for any generating set $X$ of ...
10
votes
0answers
156 views

Are all sneaky groups products of Frobenius and 2-Frobenius groups?

I've been stuck thinking about this for a while. Def. Let $G$ be a finite solvable group whose order is divisible by only three primes: $p,q,$ and $r$. Suppose that $G$ has cyclic subgroups of ...
49
votes
1answer
992 views

Are there $n$ groups of order $n$ for some $n>1$?

Given a positive integer $n$, let $N(n)$ denote the number of groups of order $n$, up to isomorphism. Question: Does $N(n)=n$ hold for some $n>1$? I checked the OEIS-sequence ...
12
votes
0answers
185 views

How does a permutation $P$ affect the singular value $\sigma_{\text{max}}(Q^\top P^\top Q)$ for orthogonal $Q$?

Let $q_i$ for $i=1,\ldots,m$ be the columns of the matrix $Q\in\mathbb{R}^{n\times m}$, $n\geq 2m$, which are pairwise orthonormal ( i.e. $q_i^\top q_j = \begin{cases} 1 & \text{if}\quad i=j \\ 0 ...
7
votes
2answers
546 views

Can all the sporadic groups be expressed as permutation groups based on a single big cycle?

Working on M11, I came up with that it can be generated using the following permutations: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10] [[2, 0, 1, 7], [3, 4, 5, 6]] [[4, 0, 6, 7], [2, 3, 1, 5]] [[0, 7], [4, 6], ...
5
votes
1answer
162 views

Strongly real elements of odd order in sporadic finite simple groups

Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution. Question: Is it true ...
16
votes
1answer
441 views

Okounkov-Vershik approach to representation theory of $S_n$

This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...
3
votes
1answer
117 views

Wild automorphisms of profinite groups

Is there a profinite group $G$, a continuous automorphism $\alpha$ of $G$ and a topologically finitely generated closed subgroup $H \leq G$ such that $\alpha(H) \lneq H$ ? Note that if an ...
1
vote
0answers
53 views

Fully residually free groups and completion

Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?
6
votes
1answer
237 views

number of maximal subgroups of the symmetric group

What is the asymptotics of the number of the maximal subgroups of $S_n$ (as a function of $n$)? This must be written down somewhere... EDIT I am actually more interested in the number of conjugacy ...
1
vote
0answers
112 views

The normalizer problem for group rings

I recently studied about The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = ...
6
votes
0answers
196 views

Cartesian square root of a measure preserving action

Let $G \curvearrowright (X,\nu)$ be probability measure preserving action of a countable discrete group. When does there exist a probability measure preserving action $G \curvearrowright (Y,\mu)$ such ...
1
vote
1answer
135 views

Determining conjugacy class of a subgroup from the sizes of its intersections with the conjugacy classes

Let $C_1,C_2,\ldots,C_n$ be the conjugacy classes of a finite group $G$. It is possible that there are two non-conjugate subgroups $K \leq G$ and $H \leq G$ such that $|H \cap C_i|=|K \cap C_i|$ for ...
4
votes
2answers
145 views

“Natural” action of a semidirect product of groups

Suppose we have a group $N$ acting on a set $X$, and a group $Q$ acting on $Y$. Moreover, we are given $\phi: Q \to \mathrm{Aut}(N)$. My question is, is there a "natural" action on a set that can be ...
5
votes
0answers
167 views

Wild automorphisms of a free group

Let $F_X$ be a free group on a countably infinite set $X$. Let $\alpha$ be an automorphism of $F_X$ and $H$ a closed subgroup of $F_X$ in the profinite topology. Is it possible that $\alpha(H) ...
6
votes
1answer
267 views

G-Correlation of Vectors

Let $\vec{a},\vec{b} \in \mathbb{R}^{n}$. Consider the function $f: S_n \to \mathbb{R}$ given by $f(\sigma):= \sum_{i=1}^{n} a_i b_{\sigma(i)}$. Let $G$ be a subgroup of $S_n$, given by $O(\log n)$ ...
2
votes
0answers
96 views

Expository papers for Feit–Thompson Theorem [duplicate]

Feit–Thompson theorem states that every finite group of odd order is solvable. Its proof is near 300 pages so it is definitely not an easy paper to read without a prior knowledge about the general ...
8
votes
1answer
201 views

Is the word problem decidable for free finitely generated self-square groups?

A self-square group is a group with extra structure, which encodes the fact that the group is isomorphic to its own direct square. To be exact, the group $G$ has a special element $1$, a unary ...
1
vote
0answers
81 views

Profinite rank of Fuchsian groups

Let $G$ be a Fuchsian group whose profinite completion is finitely generated. Must $G$ be finitely generated? A Fuchsian group is a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$ or ...
3
votes
2answers
326 views

Subgroups of hyperbolic groups

Let $G$ be a finitely generated hyperbolic group, and let $H \leq G$ be a subgroup whose profinite completion is finitely generated. Must $H$ be finitely generated? In view of Ian Agol's answer, I ...
5
votes
1answer
146 views

Upper bound on level of a congruence subgroup of the modular group

Let $\Gamma = PSL(2,\mathbb{Z}) = \langle S,T \ | \ S^2=(ST)^3=1 \rangle$. Let $G$ be some mystery normal subgroup of $\Gamma$ that we happen to think may be congruence. Recall that a subgroup of ...
14
votes
0answers
264 views

In what sense is the braid group $B_3$ the universal central extension of the modular group $\Gamma$?

First let's recall some definitions. Let $G$ be a perfect group, so that $$H^2(G, A) \cong \text{Hom}(H_2(G), A)$$ for all abelian groups $A$ by universal coefficients. This means that when $A = ...
2
votes
0answers
113 views

Centerless finite groups with exactly three prime divisors

Let $G$ be a centerless finitee group and $\pi(G)=\{p,q,r\}$ be the set of prime divisors of $|G|$. I have some questions about these groups. Is there any classification of such groups? In what ...
2
votes
1answer
120 views

Is the Frattini subgroup of a free profinite group trivial?

Let $\widehat{F_2}$ be the free profinite group of rank 2. Is its Frattini subgroup $\Phi(\widehat{F_2})$ trivial? I know that the Frattini subgroup of a pro-$p$ group is open. On the other hand, the ...
7
votes
1answer
205 views

Frucht's type theorem for Riemann surface

Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite ...
3
votes
0answers
119 views

Is there any probabilistic characterization for generalized solvable groups?

References: This question is inspired by a conjecture of Alon Amit that is solved by Miklós Abért, Nikolay Nikolov and Dan Segal in the following papers: (1) On the probability of satisfying a ...
5
votes
1answer
225 views

abelian and nonabelian parts of Aut($\widehat{F_2}$)

Let $F$ be the free profinite group on two generators. Let $\text{IA}(F) := \ker\left(\text{Aut}(F)\rightarrow GL_2(\widehat{\mathbb{Z}})\right)$, the group of "IA automorphisms" of $F$. (I'm also ...
1
vote
2answers
149 views

Abelian group of finite rank

Let given torsion free abelian group $A$ of finite rank. Let for prime number $p$, given that $\cap_i p^iA =\{0\}$. Is it true that for any $p$- torsion abelian group $B$, $\text{Hom}_{\mathbb{Z}}(A, ...
15
votes
2answers
427 views

Minimal maximal subgroup of the symmetric group

The question is pretty much in the title: What is the maximal subgroup of $S_d$ of maximal index (so minimal size)? A slight variant (I am not sure if it leads to a different answer) is: what if we ...
3
votes
1answer
177 views

An inequality for Fuchsian groups?

Let $G$ be a finitely generated Fuchsian group. (i.e. a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$). Is it true that $d(G) < 2\beta_{2}^1(G) + 1$ ? Here, $\beta_{2}^1(G)$ stands ...
5
votes
2answers
276 views

Conjugate transpose and discreteness, for Kleinian groups

Let $G=\langle g_1,\dots g_n\rangle<\mathrm{PSL}_2(\mathbb{C})$ be discrete, i.e. a finitely generated Kleinian group. Let $H=\langle g^{\dagger}g\mid g\in G\rangle$, (the group generated by the ...
0
votes
0answers
180 views

Basic question about power series and complete group algebras

This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange. Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider ...
3
votes
2answers
174 views

Automorphism group of a free product

Suppose that $G$ and $H$ are groups (not isomorphic) and $G\ast H$ the free product. Let $Aut(G)$, $Aut(H)$ be the automorphism groups of $G$ and $H$. What is $Aut(G\ast H)$ ?
2
votes
0answers
110 views

Is a finitely generated residually free group “almost LERF”?

Let $G$ be a finitely generated residually free group. (i.e. for each $1 \neq g \in G$ there exists a homomorphism $\tau \colon G \to F$ such that $F$ is a free group, and $\tau(g) \neq 1$.) ...
4
votes
1answer
77 views

Example of a doubly degenerate surface group not coming from a pseudo-Anosov mapping torus

Doubly degenerate surface groups are discrete subgroups of $PSL(2,\mathbb{C})$ whose limit set is all of $S^2$, the boundary of $\mathbb{H}^3$. A standard example of such a group is given as follows: ...
3
votes
0answers
83 views

Freeness of a matrix semigroup

Motivated by some questions in the dimension theory of self-affine sets, a colleague and I are interested in the freeness (or otherwise) of the subsemigroup of $SL_\pm(2,\mathbb{R})$ generated by the ...